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8796 TWOING INDEX

4. Greenfield, C. C. and Tam, S. M. (1976).

J. R. Statist. Soc. A, 139, 96–103.

5. Macarthur, E. W. (1983). J. R. Statist. Soc. A, 146, 85–86.

6. Nour, EI-S. (1982). J. R. Statist. Soc. A, 145, 106–116.

See also LOG-LINEARMODELS INCONTINGENCYTABLES; TETRACHORICCORRELATIONCOEFFICIENT; and TWO-BY-TWO(2× 2) TABLES.

TWOING INDEX

An index of ‘‘distinctiveness’’ or diversity∗ used in some recursive partitioning∗ proce-dures. If a node is split into two nodes, say L and R, in proportions pL: pR(pL+ pR= 1) such that the proportion of items from class (j) in L(R) is PjL(PjR), the twoing index of the split is 1 4pLpR  j|pjL− pjR| 2 .

The greater the index, the more effective the split.

See also DIVERSITYINDICESand RECURSIVEPARTITIONING.

TWO-PHASE SAMPLING.

See

SURVEYSAMPLING

TWO-SAMPLE MATCHING TEST

The two-sample matching test is a procedure to test whether two independent samples come from the same continuous distribution or not. Let X and Y be independent ran-dom variables with continuous cumulative distribution functions F and G respectively, and let X(k) and Y(k), 1 k  n, be the order

statistics∗ of independent random samples from F and G. In inference with two sam-ples, one issue of interest is whether the two samples indeed come from two different pop-ulations or not. The standard procedure is to test the null hypothesis H0: G= F.

Since comparison of two samples is a fundamental topic of statistics, interest in

testing the above hypothesis goes back to the late thirties. Smirnov [7] proposed the test statistic sup−∞<x<∞|Fn(x)− Gm(x)| for arbitrary sample sizes against the alterna-tive H1: G= F, where Fn(x) and Gm(x) are the empirical distribution functions of F and G respectively (see KOLMOGOROV–SMIRNOV

STATISTICS). The distribution of this test statistic and generalizations of it were studied later by several authors, including Tak ´acs [8], where related references can also be found.

However, if the main concern is a spe-cific alternative, other nonparametric tests with greater power are preferable. In particu-lar, the well-known nonparametric tests such as the Wilcoxon, Mann–Whitney∗, and von Mises tests are developed against location shift alternatives that assume that G(x)= F(x− ), and the corresponding alternatives H1:  > (<,=)0 then induce a stochastic

ordering between F and G [2,4].

The two-sample matching test to be described here is appropriate for any general alternative H1: F(x)= G(x). In this entry, the

exact distribution and the moments of the test statistic are presented and the limit-ing distribution is provided, together with the result that the test is consistent against a general alternative. More details are pre-sented in refs. 3, 6.

Assume without loss of generality that the distributions F and G are concentrated on the unit interval [0, 1], and that F(x)= x, 0  x  1. The test considered in Rao and Tiwari [5] and G ¨urler and Siddiqui [1] is based on the number of matches between the order statis-tics X(k)and Y(k), k= 1, . . . , n, from F and G

respectively. For the event Ak, a match occurs if X(k)∈ (Y(k−1), Y(k)] for any k, 1 k  n, with

Y(0)= 0. The matching test statistic is then

Sn= n

k=1Ik, where Ik is the indicator of Ak. Sn is a special case of a more gen-eral class of statistics considered in ref. 8. Later, Rao and Tiwari [5] provided a sim-pler proof for the special case of Sn defined above. Let P0and P1be the probability

mea-sures, and E0and E1the expectations, under

H0 and H1, respectively. Clearly, for k 1,

P0(Sn k)  P1(Sn k). Hence, for a signifi-cance level α, 0 < α < 1, the critical region for testing H0against H1is of the form Sn kn,α

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TWO-SAMPLE MATCHING TEST 8797 Table 1. X(i) Y(i) 0.094 0.0039 0.168 0.0041 0.229 0.0064 0.265 0.0116 0.384 0.0706 0.460 0.0997 0.482 0.1028 0.511 0.1069 0.523 0.5792 0.710 0.6155

where kn,α is the largest integer k such that P0(Sn k)  α.

Example. Suppose an X-sample of size 10

is observed from a uniform distribution on (0, 1), and a Y-sample of the same size is observed from a beta distribution∗ with parameters α= 0.5 and β = 1.5, with the order statistics shown in Table 1. Then Sn= 1, indicating that there is only one match-ing between the order statistics of the two samples at X(9)= 0.523, which lies between

0.1069 and 0.5792. We will see shortly that, according to the asymptotic distribution of Sn, we reject H0for this particular sample.

DISTRIBUTION, MEAN, AND VARIANCE OF

SN

An exact expression for P0(Sn k), given in ref. 8, uses lattice-path counting meth-ods. The alternative proof in ref. 5 is based on the number of crossings in the Kol-mogorov–Smirnov statistics. The exact dis-tribution of Sn under H0, for k= 1, . . . , n, is

given by P(Sn k) =  2n n+ k    2n n  . (1)

The mean and the variance of Sn, under the hypothesis H0: F= G, are given by

E[Sn]= 22n−1  2n n  −1 2, Var[Sn]= n + 1 4 − 24n−2  2n n 2.

For small n, the exact distribution of Sncan be tabulated using (1). However, when the sample size is large, it is desirable to exhibit the rejection region in a simpler form for practical purposes, via the limiting distri-bution of Sn. Under the null hypothesis of identical distributions for X and Y, and for x > 0 [1],

lim

n→∞P0(n−1/2Sn x) = e−x

2

.

From this result, approximate critical regions of the test can be found. For a size-α test, we reject if Sn kn,α, where kn,α= [−n ln(1 − α)]1/2. Then, if we fix α= 0.05, for n = 5, 10, 15, 20, 30, the kn,α-values are obtained as kn,0.05= 0.506, 0.716, 0.877, 1.01, 1.24 respectively. In the example above, since S10= 1 > 0.716, we reject H0.

Consistency

A test is consistent if its power tends to one as n tends to infinity, against a fixed alternative. If the measure of the interval on which F(x) and G(x) do not cross each other is one (that is, if they cross each other only at countably many points), then the test based on Snis consistent for any significance level α, 0 < α < 1 [1]. This result guaran-tees that as the sample size increases, the power of the matching test converges to one against a general alternative and therefore the test is consistent. However, the result does not indicate anything about the small-sample behavior of the test and the rate of convergence for the power.

REFERENCES

1. G ¨urler, ¨U. and Siddiqui, M. M. (1996). On the consistency of a two-sample matching test.

Non-parametric Statist., 7, 69–73.

2. H ´ajek, J. and ˇSid ´ak, Z. (1967). Theory of Ranked

Tests. Academic Press, New York.

3. Khidr, A. M. (1981). Matching of order statis-tics with intervals. Indian J. Pure Appl. Math., 12, 1402–1407.

4. Randles, R. H. and Wolfe, D. A. (1979).

Intro-duction to the Theory of Nonparametric Statis-tics. Wiley, New York.

5. Rao, J. S. and Tiwari, R. C. (1983). One- and two-sample match statistics. Statist. Probab.

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8798 TWO-SAMPLE PROBLEM

6. Siddiqui, M. M. (1982). The consistency of a matching test. J. Statist. Plann. Inference, 6, 227–233.

7. Smirnov, N. V. (1939). On the estimation of the discrepancy between empirical curves of distribution for two independent samples. (In Russian.) Bull. Math. Univ. Moscow A, 2, 3–14. 8. Tak ´acs, L. (1971). On the comparison of two empirical distribution functions. Ann. Math.

Statist., 42, 1157–1166.

See also EMPIRICALDISTRIBUTIONFUNCTION(EDF) STATISTICS; GOODNESS-OF-FITDISTRIBUTION, TAK ´ACS; KOLMOGOROV–SMIRNOVSTATISTICS; and ORDER STATISTICS.

¨

ULK ¨UGURLER¨

TWO-SAMPLE PROBLEM.

See

LOCATIONTESTS

TWO-SAMPLE PROBLEM,

BAUM-GARTNER–WEISS–SCHINDLER

TEST

Tests that two independent samples

X1, . . . , Xmand Y1, . . . , Ymbelong to the same continuous population (two-sample problem) abound in the statistical literature. Among them, the Kolmogorov-Smirnov* test, the Cram´er-von Mises* test, and the Mann-Whitney–Wilcoxon* test are perhaps the most popular in applications.

These tests are based on comparisons of the two empirical* distribution functions or edfs Fn(x)=    0 for x < X1, i/n for Xi< x < Xi+1 1 for x > Xn,

and Fm(x), defined analogously for the sam-ple Y1, Y2, . . . , Ym. (Observations here are ordered in increasing order of magnitude.)

The Kolmogorov-Smirnov test uses the maximum of |Fn(x)− Fm(x)|, the Wilcoxon test utilizes the integral of [Fn(x)− Fm(x)], and the Cram´er-von Mises test employs the squared norm of [Fn(x)− Fm(x)] as test val-ues.

The Baumgartner–Weiss–Schindler

(BWS) test [2] uses the squared norm of the difference [Fn(x)− Fm(x)] weighted by

its variance, an idea borrowed from the Anderson-Darling test for the one-sample problem [1].

Specifically, for testing the null hypothesis that both samples are drawn from the same population with continuous but unknown cdf F(x), the integral transformation

Z= F(X)

is introduced, which results in Z being almost surely uniformly distributed on [0,1] if X has the cdf F(x). Applying this transformation to the two sets of data, we arrive at the edfs

ˆ

Fn(z) and ˆFm(z). The difference ˆFn(z)− ˆFm(z) is weighted by [z(1− z)]−1, yielding a test value ˜B = mn m+ n  1 0 [z(1− z)]−1( ˆFn(z)− ˆFm(z))2dz (1) Since F(x) is unknown, BWS 2 propose to approximate Equation (1) using ranks. The rank Gi (resp. Hi) of each element Xi (resp. Yj) is defined as the number of data values in both samples smaller than or equal to Xi (resp. Yj). Then ˜B is approximated by

BX = 1 n n  i=1 (Gim+nn i)2 i n+1(1− i n+1) m(m+n) n (2) or by BY= 1 m m  j=1 (Hjmm+nj)2 j m+1(1−m+1j )n(mm+n) (3)

and new test statistic B is defined to be B= (BX+ BY)/2.

The processes in Equations (2) and (3) converge to a standard Brownian bridge* as n, m→ ∞ and n/m → a (where a is a finite constant). Applying the Anderson-Darling technique [1], BWS derive the asymptotic distribution of B to be (b)= lim n,m→∞Pr(B < b) =  π 2 1 b ∞  j=0  −1 2 j  (4j+ 1)  1 0 1  r3(1− r) × exp  rb 8 − π2(4j+ 1)2 8rb  dr  ,

Referanslar

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